2

I have an integral problem as follows: $$ I(k,\sigma)=\int_{-\infty}^{\infty}\frac{\exp\left(-\frac{x^2}{2\sigma^2}\right)}{\sqrt{1+k^2\sin^2(x)}}dx, $$ where $k,\sigma>0$. The function to be integrated is an elliptic function multiplied by a Gaussian kernel. I searched from Google and have no solution. Maybe one possible approach is first to find the power series of the elliptic function: $$ \frac{1}{\sqrt{1+k^2\sin^2(x)}}=\sum_{n=0}^{\infty}a_n(k)x^n. $$ However, this is not easy too.

Jan Eerland
  • 28,671
Mr.qin
  • 21
  • 1
    Assuming $|k|<1$$$\frac{1}{\sqrt{1+k^2\sin^2(x)}}=\sum_{n\geq 0}\frac{(-1)^n}{4^n}\binom{2n}{n}k^{2n}\sin(x)^{2n}$$ hence you just have to compute $$ I_n = \int_{\mathbb{R}}\sin(x)^{2n}\exp\left(-\frac{x^2}{2\sigma^2}\right),dx$$ – Jack D'Aurizio Aug 14 '22 at 08:08
  • First, thank you so much. I have tried your idea. According to the Power-reduction formula, one can get the result of $I_n$ with the form of infinite series summation. Finally, $I(k,\sigma)$ could be written as a summation of multiple infinite series. Is it possible to simplify the result further? – Mr.qin Aug 15 '22 at 14:03

0 Answers0